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Towards the Albertson Conjecture

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Towards the Albertson Conjecture Towards the Albertson conjecture J´anos Bar´at∗ Department of Computer Science and Systems Technology University of Pannonia, Egyetem u. 10, 8200 Veszpr´em, Hungary [email protected] G´eza T´oth† R´enyi Institute, Re´altanoda u. 13-15, 1052 Budapest, Hungary [email protected] Submitted: Sep 2, 2009; Accepted: May 7, 2010; Published: May 14, 2010 Mathematics Subject Classification: 05C10, 05C15 Abstract Albertson conjectured that if a graph G has chromatic number r, then the crossing number of G is at least as large as the crossing number of Kr, the complete graph on r vertices. Albertson, Cranston, and Fox verified the conjecture for r 6 12. In this paper we prove it for r 6 16. Dedicated to the memory of Michael O. Albertson. 1 Introduction Graphs in this paper are without loops and multiple edges. Every planar graph is four- colorable by the Four Color Theorem [2, 24]. The efforts to solve the Four Color Problem had a great effect on the development of graph theory, and FCT is one of the most important theorems of the field. The crossing number of a graph G, denoted cr(G), is the minimum number of edge crossings in a drawing of G in the plane. It is a natural relaxation of planarity, see [25] for a survey. The chromatic number of a graph G, denoted χ(G), is the minimum number of colors in a proper coloring of G. The Four Color Theorem states: if cr(G) = 0, then χ(G) 6 4. Oporowski and Zhao [18] proved that every graph with crossing number at most two is 5-colorable. Albertson et al. [5] showed that if cr(G) 6 6, then χ(G) 6 6. It ∗Research is supported by OTKA Grants PD 75837 and K 76099, and the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences. †Research is supported by OTKA T 038397 and T 046246. the electronic journal of combinatorics 17 (2010), #R73 1 was observed by Schaefer that if cr(G)= k, then χ(G)= O(√4 k), and this is the correct order of magnitude [4]. Graphs with chromatic number r do not necessarily contain Kr as a subgraph, they can have clique number 2, see [27]. The Haj´os conjecture proposed that graphs with chromatic number r contain a subdivision of Kr. This conjecture, whose origin is unclear but attributed to Haj´os, turned out to be false for r > 7. Also, it was shown by Erd˝os and Fajtlowicz [9] that almost all graphs are counterexamples. Albertson posed the following Conjecture 1 If χ(G)= r, then cr(G) > cr(Kr). This statement is weaker than Haj´os’ conjecture: if G contains a subdivision of Kr, then cr(G) > cr(Kr). For r = 5, Albertson’s conjecture is equivalent to the Four Color Theorem. Oporowski and Zhao [18] verified it for r = 6. Albertson, Cranston, and Fox [4] proved it for r 6 12. In this note, we take one more little step. Theorem 2 For r 6 16, if χ(G)= r, then cr(G) > cr(Kr). In their proof, Albertson, Cranston and Fox combined lower bounds for the number of edges of r-critical graphs, and lower bounds on the crossing number of graphs with given number of vertices and edges. Our proof is very similar, but we use better lower bounds in both cases. Albertson et al. proved that any minimal counterexample to Conjecture 1 should have less than 4r vertices. We slightly improve this result as follows. Lemma 3 If G is an n-vertex, r-critical graph with n > 3.57r, then cr(G) > cr(Kr). In Section 2, we review lower bounds for the number of edges of r-critical graphs. In Section 3, we discuss lower bounds on the crossing number. In Section 4, we combine these two bounds to obtain the proof of Theorem 2. In Section 5, we prove Lemma 3. Let n always denote the number of vertices of G. In notation and terminology, we follow Bondy and Murty [6]. In particular, the join of two disjoint graphs G and H, denoted G H, arises by adding all edges between vertices of G and H. A vertex v is of full degree∨, if it has degree n 1. If a graph G contains a subdivision of H, then G contains a topological H. A vertex− v is adjacent to a vertex set X means that each vertex of X is adjacent to v. 2 Color-critical graphs A graph G is r-critical, if χ(G)= r, but all proper subgraphs of G have chromatic number less than r. In what follows, let G denote an r-critical graph with n vertices and m edges. Since G is r-critical, every vertex has degree at least r 1, therefore, 2m > (r 1)n. The − − value 2m (r 1)n is the excess of G. For r > 3, Dirac [7] proved the following: if G is not complete,− then− 2m > (r 1)n +(r 3). For r > 4, Dirac [8] gave a characterization − − the electronic journal of combinatorics 17 (2010), #R73 2 of r-critical graphs with excess r 3. For a positive integer r, r > 3, let ∆r be the following family of graphs. For any− graph in the family, let the vertex set consist of three non-empty, pairwise disjoint sets A, B1, B2 and two additional vertices a and b. Here, B + B = A +1= r 1. The sets A and B B both span cliques, a is connected to | 1| | 2| | | − 1 ∪ 2 A B and b is connected to A B . See Figure 1. Graphs in ∆r are called Haj´os graphs ∪ 1 ∪ 2 of order 2r 1. Observe, that these graphs have chromatic number r, and they contain − a topological Kr. Hence they satisfy Haj´os’ conjecture. a B1 A B2 b Figure 1: The family ∆r Gallai [10] proved that any r-critical graph with at most 2r 2 vertices is the join of − two smaller graphs. Therefore, the complement of any such graph is disconnected. Based on this observation, Gallai proved that non-complete r-critical graphs on at most 2r 2 − vertices have much larger excess than in Dirac’s result. Lemma 4 [10] Let r, p be integers, r > 4 and 2 6 p 6 r 1. If G is an r-critical graph with n vertices and m edges, where n = r + p, then 2m > (−r 1)n + p(r p) 2. Equality − − − holds if and only if G is the join of Kr−p− and G ∆p . 1 ∈ +1 Since every G in ∆p+1 contains a topological Kp+1, the join of Kr−p−1 and G contains a topological Kr. This yields a slight improvement for our purposes. Corollary 5 Let r, p be integers, r > 4 and 2 6 p 6 r 1. If G is an r-critical graph − with n vertices and m edges, where n = r + p, and G does not contain a topological Kr, then 2m > (r 1)n + p(r p) 1. − − − the electronic journal of combinatorics 17 (2010), #R73 3 We call the bound given by Corollary 5 the Gallai bound. For r > 3, let r denote the family of the following graphs G. The vertex set of E any G consists of four non-empty pairwise disjoint sets A1, A2, B1, B2, and one additional vertex c. Here B1 + B2 = A1 + A2 = r 1 and A2 + B2 6 r 1. Let A = A1 A2 and B = B B| .| The| sets| A| and| B| each| induce− a clique| | in| G.| The− vertex c is connected∪ 1 ∪ 2 to A B . A vertex a in A is adjacent to a vertex b in B if and only if a A and b B . 1 ∪ 1 ∈ 2 ∈ 2 c A1 B1 A2 B2 Figure 2: The family r E Observe, that r ∆r, and every graph G in r is r-critical with 2r 1 vertices. E ⊃ E − Kostochka and Stiebitz [15] improved Dirac’s bound as follows. Lemma 6 [15] Let r be a positive integer, r > 4, and let G be an r-critical graph. If G is neither Kr nor a member of r, then 2m > (r 1)n + (2r 6). E − − Corollary 7 Let r be a positive integer, r > 4, and let G be an r-critical graph. If G does not contain a topological Kr, then 2m > (r 1)n + (2r 6). − − Proof: We show that any member of r contains a topological Kr. The sets A and B E both span a complete graph on r 1 vertices. We only have to show that vertex c is − connected to A2 or B2 by vertex-disjoint paths. To see this, we observe that A2 or B2 is the smallest of A , A , B , B . Indeed, if B was the smallest, then| A | > |B | {| 1| | 2| | 1| | 2|} | 1| | 2| | 1| implies A2 + B2 > B1 + B2 = r 1 contradicting our assumption. We may assume that A | is| the| smallest.| | | Now| c| is adjacent− to A , and there is a matching of size A | 2| 1 | 2| between B1 and B2 and between B2 and A2. Therefore, we can find a set S of disjoint paths from c to A . In this way, A c S is a topological r-clique. 2 2 ∪ ∪ The bound in Corollary 7 is the Kostochka, Stiebitz bound, or KS-bound for short. In what follows, we obtain a complete characterization of r-critical graphs on r +3 or r + 4 vertices.
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